ELLIPRG

Carlson’s symmetric integral R_G is a three-argument second-kind symmetric form that complements R_F and R_D in elliptic-integral identities.

R_G(x,y,z)=\frac{1}{4}\int_0^{\infty}[(t+x)(t+y)(t+z)]^{-1/2}\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\,dt

This wrapper evaluates R_G for scalar real inputs.

Excel Usage

=ELLIPRG(x, y, z)

x (float, required): First nonnegative real parameter.
y (float, required): Second nonnegative real parameter.
z (float, required): Third nonnegative real parameter.

Returns (float): Value of Carlson completely symmetric elliptic integral RG.

Example 1: Carlson RG with positive parameters

Inputs:

x	y	z
1	2	3

Excel formula:

=ELLIPRG(1, 2, 3)

Expected output:

1.40185

Example 2: Carlson RG with one zero parameter

Inputs:

x	y	z
0	2	2

Excel formula:

=ELLIPRG(0, 2, 2)

Expected output:

1.11072

Example 3: Carlson RG when all parameters are equal

Inputs:

x	y	z
2	2	2

Excel formula:

=ELLIPRG(2, 2, 2)

Expected output:

1.41421

Example 4: Carlson RG with fractional parameters

Inputs:

x	y	z
0.5	1.5	2.5

Excel formula:

=ELLIPRG(0.5, 1.5, 2.5)

Expected output:

1.20486

Python Code

from scipy.special import elliprg as scipy_elliprg

def elliprg(x, y, z):
    """
    Compute Carlson's completely symmetric elliptic integral RG.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.elliprg.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        x (float): First nonnegative real parameter.
        y (float): Second nonnegative real parameter.
        z (float): Third nonnegative real parameter.

    Returns:
        float: Value of Carlson completely symmetric elliptic integral RG.
    """
    try:
        return float(scipy_elliprg(x, y, z))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

x *

First nonnegative real parameter.

y *

Second nonnegative real parameter.

z *

Third nonnegative real parameter.